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Properties

Label 2-2e9-32.21-c1-0-5
Degree $2$
Conductor $512$
Sign $0.788 - 0.615i$
Analytic cond. $4.08834$
Root an. cond. $2.02196$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.191 + 0.0794i)3-s + (0.292 − 0.707i)5-s + (2.27 + 2.27i)7-s + (−2.09 + 2.09i)9-s + (3.60 + 1.49i)11-s + (−1.86 − 4.50i)13-s + 0.158i·15-s + 3.05i·17-s + (1.60 + 3.87i)19-s + (−0.616 − 0.255i)21-s + (−0.271 + 0.271i)23-s + (3.12 + 3.12i)25-s + (0.473 − 1.14i)27-s + (2.24 − 0.931i)29-s + 6.82·31-s + ⋯
L(s)  = 1  + (−0.110 + 0.0458i)3-s + (0.130 − 0.316i)5-s + (0.858 + 0.858i)7-s + (−0.696 + 0.696i)9-s + (1.08 + 0.450i)11-s + (−0.517 − 1.24i)13-s + 0.0410i·15-s + 0.740i·17-s + (0.368 + 0.889i)19-s + (−0.134 − 0.0557i)21-s + (−0.0565 + 0.0565i)23-s + (0.624 + 0.624i)25-s + (0.0911 − 0.219i)27-s + (0.417 − 0.173i)29-s + 1.22·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.788 - 0.615i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.788 - 0.615i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(512\)    =    \(2^{9}\)
Sign: $0.788 - 0.615i$
Analytic conductor: \(4.08834\)
Root analytic conductor: \(2.02196\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{512} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 512,\ (\ :1/2),\ 0.788 - 0.615i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.42743 + 0.490956i\)
\(L(\frac12)\) \(\approx\) \(1.42743 + 0.490956i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
good3 \( 1 + (0.191 - 0.0794i)T + (2.12 - 2.12i)T^{2} \)
5 \( 1 + (-0.292 + 0.707i)T + (-3.53 - 3.53i)T^{2} \)
7 \( 1 + (-2.27 - 2.27i)T + 7iT^{2} \)
11 \( 1 + (-3.60 - 1.49i)T + (7.77 + 7.77i)T^{2} \)
13 \( 1 + (1.86 + 4.50i)T + (-9.19 + 9.19i)T^{2} \)
17 \( 1 - 3.05iT - 17T^{2} \)
19 \( 1 + (-1.60 - 3.87i)T + (-13.4 + 13.4i)T^{2} \)
23 \( 1 + (0.271 - 0.271i)T - 23iT^{2} \)
29 \( 1 + (-2.24 + 0.931i)T + (20.5 - 20.5i)T^{2} \)
31 \( 1 - 6.82T + 31T^{2} \)
37 \( 1 + (-1.50 + 3.63i)T + (-26.1 - 26.1i)T^{2} \)
41 \( 1 + (-1.54 + 1.54i)T - 41iT^{2} \)
43 \( 1 + (1.80 + 0.748i)T + (30.4 + 30.4i)T^{2} \)
47 \( 1 - 7.37iT - 47T^{2} \)
53 \( 1 + (4.04 + 1.67i)T + (37.4 + 37.4i)T^{2} \)
59 \( 1 + (4.19 - 10.1i)T + (-41.7 - 41.7i)T^{2} \)
61 \( 1 + (3.28 - 1.35i)T + (43.1 - 43.1i)T^{2} \)
67 \( 1 + (-4.81 + 1.99i)T + (47.3 - 47.3i)T^{2} \)
71 \( 1 + (6.47 + 6.47i)T + 71iT^{2} \)
73 \( 1 + (-2.84 + 2.84i)T - 73iT^{2} \)
79 \( 1 + 9.74iT - 79T^{2} \)
83 \( 1 + (3.74 + 9.04i)T + (-58.6 + 58.6i)T^{2} \)
89 \( 1 + (7.58 + 7.58i)T + 89iT^{2} \)
97 \( 1 - 3.71T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.01342831861041373151293440990, −10.16175384611672119433583335530, −9.107139001788601773808375181709, −8.297376968319935240858362667294, −7.63292807988147334343373984679, −6.11791835041408663237280919512, −5.39398549953077912970037642712, −4.49528751850476131014802070235, −2.91519779611334118898603250632, −1.58986431222424686484865740674, 1.05561598336297316492855715690, 2.78246256488193253827444013473, 4.12034024001927669804832613771, 4.99730435741539223481813096880, 6.53735871528308613475071568536, 6.86251531280307871476106230842, 8.170961340732710281683821886757, 9.075469827901232834068763031916, 9.838031352589038449485980510795, 11.07793694361045542000011277861

Graph of the $Z$-function along the critical line